3.1992 \(\int \frac{(d+e x)^{3/2}}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx\)

Optimal. Leaf size=83 \[ \frac{2 \sqrt{d+e x}}{c d}-\frac{2 \sqrt{c d^2-a e^2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{c^{3/2} d^{3/2}} \]

[Out]

(2*Sqrt[d + e*x])/(c*d) - (2*Sqrt[c*d^2 - a*e^2]*ArcTanh[(Sqrt[c]*Sqrt[d]*Sqrt[d
+ e*x])/Sqrt[c*d^2 - a*e^2]])/(c^(3/2)*d^(3/2))

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Rubi [A]  time = 0.139233, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.108 \[ \frac{2 \sqrt{d+e x}}{c d}-\frac{2 \sqrt{c d^2-a e^2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{c^{3/2} d^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^(3/2)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2),x]

[Out]

(2*Sqrt[d + e*x])/(c*d) - (2*Sqrt[c*d^2 - a*e^2]*ArcTanh[(Sqrt[c]*Sqrt[d]*Sqrt[d
+ e*x])/Sqrt[c*d^2 - a*e^2]])/(c^(3/2)*d^(3/2))

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Rubi in Sympy [A]  time = 39.4042, size = 71, normalized size = 0.86 \[ \frac{2 \sqrt{d + e x}}{c d} - \frac{2 \sqrt{a e^{2} - c d^{2}} \operatorname{atan}{\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d + e x}}{\sqrt{a e^{2} - c d^{2}}} \right )}}{c^{\frac{3}{2}} d^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**(3/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2),x)

[Out]

2*sqrt(d + e*x)/(c*d) - 2*sqrt(a*e**2 - c*d**2)*atan(sqrt(c)*sqrt(d)*sqrt(d + e*
x)/sqrt(a*e**2 - c*d**2))/(c**(3/2)*d**(3/2))

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Mathematica [A]  time = 0.0713671, size = 83, normalized size = 1. \[ \frac{2 \sqrt{d+e x}}{c d}-\frac{2 \sqrt{c d^2-a e^2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{c^{3/2} d^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^(3/2)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2),x]

[Out]

(2*Sqrt[d + e*x])/(c*d) - (2*Sqrt[c*d^2 - a*e^2]*ArcTanh[(Sqrt[c]*Sqrt[d]*Sqrt[d
+ e*x])/Sqrt[c*d^2 - a*e^2]])/(c^(3/2)*d^(3/2))

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Maple [A]  time = 0.01, size = 122, normalized size = 1.5 \[ 2\,{\frac{\sqrt{ex+d}}{cd}}-2\,{\frac{a{e}^{2}}{cd\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}\arctan \left ({\frac{cd\sqrt{ex+d}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}} \right ) }+2\,{\frac{d}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}\arctan \left ({\frac{cd\sqrt{ex+d}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^(3/2)/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2),x)

[Out]

2*(e*x+d)^(1/2)/c/d-2/c/d/((a*e^2-c*d^2)*c*d)^(1/2)*arctan(c*d*(e*x+d)^(1/2)/((a
*e^2-c*d^2)*c*d)^(1/2))*a*e^2+2*d/((a*e^2-c*d^2)*c*d)^(1/2)*arctan(c*d*(e*x+d)^(
1/2)/((a*e^2-c*d^2)*c*d)^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(3/2)/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.245094, size = 1, normalized size = 0.01 \[ \left [\frac{\sqrt{\frac{c d^{2} - a e^{2}}{c d}} \log \left (\frac{c d e x + 2 \, c d^{2} - a e^{2} - 2 \, \sqrt{e x + d} c d \sqrt{\frac{c d^{2} - a e^{2}}{c d}}}{c d x + a e}\right ) + 2 \, \sqrt{e x + d}}{c d}, -\frac{2 \,{\left (\sqrt{-\frac{c d^{2} - a e^{2}}{c d}} \arctan \left (\frac{\sqrt{e x + d}}{\sqrt{-\frac{c d^{2} - a e^{2}}{c d}}}\right ) - \sqrt{e x + d}\right )}}{c d}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(3/2)/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x),x, algorithm="fricas")

[Out]

[(sqrt((c*d^2 - a*e^2)/(c*d))*log((c*d*e*x + 2*c*d^2 - a*e^2 - 2*sqrt(e*x + d)*c
*d*sqrt((c*d^2 - a*e^2)/(c*d)))/(c*d*x + a*e)) + 2*sqrt(e*x + d))/(c*d), -2*(sqr
t(-(c*d^2 - a*e^2)/(c*d))*arctan(sqrt(e*x + d)/sqrt(-(c*d^2 - a*e^2)/(c*d))) - s
qrt(e*x + d))/(c*d)]

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Sympy [A]  time = 9.94536, size = 246, normalized size = 2.96 \[ \frac{2 \left (\frac{e \sqrt{d + e x}}{c d} - \frac{e \left (a e^{2} - c d^{2}\right ) \left (\begin{cases} \frac{\operatorname{atan}{\left (\frac{\sqrt{d + e x}}{\sqrt{\frac{a e^{2} - c d^{2}}{c d}}} \right )}}{c d \sqrt{\frac{a e^{2} - c d^{2}}{c d}}} & \text{for}\: \frac{a e^{2} - c d^{2}}{c d} > 0 \\- \frac{\operatorname{acoth}{\left (\frac{\sqrt{d + e x}}{\sqrt{\frac{- a e^{2} + c d^{2}}{c d}}} \right )}}{c d \sqrt{\frac{- a e^{2} + c d^{2}}{c d}}} & \text{for}\: d + e x > \frac{- a e^{2} + c d^{2}}{c d} \wedge \frac{a e^{2} - c d^{2}}{c d} < 0 \\- \frac{\operatorname{atanh}{\left (\frac{\sqrt{d + e x}}{\sqrt{\frac{- a e^{2} + c d^{2}}{c d}}} \right )}}{c d \sqrt{\frac{- a e^{2} + c d^{2}}{c d}}} & \text{for}\: \frac{a e^{2} - c d^{2}}{c d} < 0 \wedge d + e x < \frac{- a e^{2} + c d^{2}}{c d} \end{cases}\right )}{c d}\right )}{e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**(3/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2),x)

[Out]

2*(e*sqrt(d + e*x)/(c*d) - e*(a*e**2 - c*d**2)*Piecewise((atan(sqrt(d + e*x)/sqr
t((a*e**2 - c*d**2)/(c*d)))/(c*d*sqrt((a*e**2 - c*d**2)/(c*d))), (a*e**2 - c*d**
2)/(c*d) > 0), (-acoth(sqrt(d + e*x)/sqrt((-a*e**2 + c*d**2)/(c*d)))/(c*d*sqrt((
-a*e**2 + c*d**2)/(c*d))), ((a*e**2 - c*d**2)/(c*d) < 0) & (d + e*x > (-a*e**2 +
c*d**2)/(c*d))), (-atanh(sqrt(d + e*x)/sqrt((-a*e**2 + c*d**2)/(c*d)))/(c*d*sqr
t((-a*e**2 + c*d**2)/(c*d))), ((a*e**2 - c*d**2)/(c*d) < 0) & (d + e*x < (-a*e**
2 + c*d**2)/(c*d))))/(c*d))/e

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GIAC/XCAS [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(3/2)/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x),x, algorithm="giac")

[Out]

Timed out